Problem 72 Combine like terms. $$ 0.004... [FREE SOLUTION]

Chapter 0: Problem 72

Combine like terms. $$ 0.004 m^{4} n-0.005 m^{3} n^{2}-0.01 m^{4} n+0.007 m^{3} n^{2} $$

Short Answer

Expert verified

-0.006 m^{4} n + 0.002 m^{3} n^{2}

Step by step solution

Identify Like Terms

Look at the expression and group terms that have the same variables raised to the same powers. In this case, group together terms with the same combination of powers of m and n: 0.004 m^{4} n and -0.01 m^{4} n, also -0.005 m^{3} n^{2} and 0.007 m^{3} n^{2}.

Combine Like Terms for m^{4} n

Add the coefficients of the terms with m^{4}n: 0.004 - 0.01 = -0.006 so 0.004 m^{4} n - 0.01 m^{4} n = -0.006 m^{4} n.

Combine Like Terms for m^{3} n^{2}

Add the coefficients of the terms with m^{3} n^{2}: -0.005 + 0.007 = 0.002 so -0.005 m^{3} n^{2} + 0.007 m^{3} n^{2} = 0.002 m^{3} n^{2}.

Write the Final Expression

Combine the simplified terms to get the final expression: -0.006 m^{4} n + 0.002 m^{3} n^{2}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Simplification

Polynomial simplification refers to the process of reducing a polynomial to its simplest form by combining like terms and eliminating any unnecessary parts. This makes the polynomial easier to understand and work with.

In the given exercise, we start by identifying the like terms in the expression. Only terms with the exact same variable components raised to the same powers can be combined.

For example, in the exercise:

Terms with variable combination $m^4n$: 0.004 m^{4} n and -0.01 m^{4} n.
Terms with variable combination $m^3n^2$: -0.005 m^{3} n^{2} and 0.007 m^{3} n^{2}.

Once we identify the like terms, we combine them by adding or subtracting their coefficients. Let's see how this works in detail.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In our exercise, the given algebraic expression is: $$0.004 m^{4} n - 0.005 m^{3} n^{2} - 0.01 m^{4} n + 0.007 m^{3} n^{2}.$$

The key to simplifying algebraic expressions is to understand that like terms鈥攖erms that have the same variables and exponents鈥攃an be combined. This involves recognizing patterns and grouping these terms together.

For instance:

The terms $0.004 m^{4} n$ and $-0.01 m^{4} n$ can be grouped because they both have the variable $m^4n$.
Similarly, $-0.005 m^{3} n^{2}$ and $0.007 m^{3} n^{2}$ can be grouped since they both contain the variable $m^3n^2$.

Combining these grouped terms helps in reducing the expression to its simplest form.

Coefficient Addition

Coefficient addition is the process of adding or subtracting the numerical parts of like terms in an algebraic expression. This step is crucial for combining like terms and simplifying the overall expression.

In our exercise, we need to combine the coefficients of the like terms:

For $m^4n$ terms: $0.004 - 0.01 = -0.006$
For $m^3n^2$ terms: $-0.005 + 0.007 = 0.002$

Once the coefficients are added or subtracted, we multiply the resulting coefficient by the common variable part to get the simplified term. For the exercise, the combined terms result in the final expression: $$-0.006 m^{4} n + 0.002 m^{3} n^{2}.$$

This simplified expression is much easier to work with and better reflects the combined magnitude and direction of the original terms.

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91影视

Combine like terms. $$ 0.004 m^{4} n-0.005 m^{3} n^{2}-0.01 m^{4} n+0.007 m^{3} n^{2} $$

Short Answer

Step by step solution

Identify Like Terms

Combine Like Terms for m^{4} n

Combine Like Terms for m^{3} n^{2}

Write the Final Expression

Key Concepts

Polynomial Simplification

Algebraic Expressions

Coefficient Addition

One App. One Place for Learning.

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